Editing Abelian and Tauberian theorems

{{Short description|Used in the summation of divergent series}}
In [[mathematics]], '''Abelian and Tauberian theorems''' are [[theorem]]s giving conditions for two methods of summing [[divergent series]] to give the same result, named after [[Niels Henrik Abel]] and [[Alfred Tauber]]. The original examples are [[Abel's theorem]] showing that if a series [[convergent series|converges]] to some limit then its [[Abel sum]] is the same limit, and Tauber's theorem showing that if the Abel sum of a series exists and the coefficients are sufficiently small (o(1/''n'')) then the series converges to the Abel sum. More general Abelian and Tauberian theorems give similar results for more general summation methods.

There is not yet a clear distinction between Abelian and Tauberian theorems, and no generally accepted definition of what these terms mean. Often, a theorem is called "Abelian" if it shows that some summation method gives the usual sum for convergent series, and is called "Tauberian" if it gives conditions for a series summable by some method that allows it to be summable in the usual sense.

In the theory of [[integral transform]]s, Abelian theorems give the asymptotic behaviour of the transform based on properties of the original function. Conversely, Tauberian theorems give the asymptotic behaviour of the original function based on properties of the transform but usually require some restrictions on the original function.<ref>{{Cite thesis|last=Froese Fischer|first=Charlotte|date=1954|title=A method for finding the asymptotic behavior of a function from its Laplace transform|publisher=University of British Columbia |language=en|doi=10.14288/1.0080631}}</ref>

==Abelian theorems==

For any summation method ''L'', its '''Abelian theorem''' is the result that if ''c'' = (''c''<sub>''n''</sub>) is a [[Limit of a sequence|convergent sequence]], with [[Limit of a sequence|limit]] ''C'', then ''L''(''c'') = ''C''. {{clarify|date=March 2022}}

An example is given by the [[Cesàro mean|Cesàro method]], in which ''L'' is defined as the limit of the [[arithmetic mean]]s of the first ''N'' terms of ''c'', as ''N'' tends to infinity. One can [[mathematical proof|prove]] that if ''c'' does converge to ''C'', then so does the sequence (''d''<sub>''N''</sub>) where

: <math>d_N = \frac{c_1+c_2+\cdots+c_N} N.</math>

To see that, subtract ''C'' everywhere to reduce to the case ''C'' = 0. Then divide the sequence into an initial segment, and a tail of small terms: given any ε > 0 we can take ''N'' large enough to make the initial segment of terms up to ''c''<sub>''N''</sub> average to at most ''ε''/2, while each term in the tail is bounded by ε/2 so that the average is also necessarily  bounded.

The name derives from [[Abel's theorem]] on [[power series]]. In that case ''L'' is the ''radial limit'' (thought of within the [[complex plane|complex]] [[unit disk]]), where we let ''r'' tend to the limit 1 from below along the real axis in the power series with term

: ''a''<sub>''n''</sub>''z''<sup>''n''</sup>

and set ''z'' = ''r'' ·''e''<sup>''iθ''</sup>. That theorem has its main interest in the case that the power series has [[radius of convergence]] exactly 1: if the radius of convergence is greater than one, the convergence of the power series is [[uniform convergence|uniform]] for ''r'' in [0,1] so that the sum is automatically [[continuous function|continuous]] and it follows directly that the limit as ''r'' tends up to 1 is simply the sum of the ''a''<sub>''n''</sub>. When the radius is 1 the power series will have some singularity on |''z''| = 1; the assertion is that, nonetheless, if the sum of the ''a''<sub>''n''</sub> exists, it is equal to the limit over ''r''. This therefore fits exactly into the abstract picture.

==Tauberian theorems==

Partial [[converse (logic)|converses]] to Abelian theorems are called '''Tauberian theorems'''. The original result of {{harvs|txt|authorlink=Alfred Tauber|last=Tauber|first=Alfred|year=1897}}<ref>{{cite journal | last=Tauber | first=Alfred | author-link=Alfred Tauber | title=Ein Satz aus der Theorie der unendlichen Reihen|trans-title=A theorem about infinite series | language=German | doi=10.1007/BF01696278 | year=1897 | journal=[[Monatshefte für Mathematik und Physik]]| volume=8 | pages=273–277 |url=http://www.literature.at/viewer.alo?viewmode=overview&olfullscreen=true&objid=12409&page=280 | jfm=28.0221.02| s2cid=120692627 }}</ref> stated that if we assume also

:''a''<sub>''n''</sub> = o(1/''n'')

(see [[Big O notation#Little-o notation|Little o notation]]) and the radial limit exists, then the series obtained by setting ''z'' = 1 is actually convergent. This was strengthened by [[John Edensor Littlewood]]: we need only assume O(1/''n'').  A sweeping generalization is the [[Hardy–Littlewood Tauberian theorem]].

In the abstract setting, therefore, an ''Abelian'' theorem states that the domain of ''L'' contains the convergent sequences, and its values there are equal to those of the ''Lim'' functional. A ''Tauberian'' theorem states, under some growth condition, that the domain of ''L'' is exactly the convergent sequences and no more.

If one thinks of ''L'' as some generalised type of ''weighted average'', taken to the limit, a Tauberian theorem allows one to discard the weighting, under the correct hypotheses. There are many applications of this kind of result in [[number theory]], in particular in handling [[Dirichlet series]].

The development of the field of Tauberian theorems received a fresh turn with [[Norbert Wiener]]'s very general results, namely [[Wiener's Tauberian theorem]] and its large collection of [[corollaries]].<ref>{{cite journal | first=Norbert |last=Wiener | authorlink=Norbert Wiener | title=Tauberian theorems |  year=1932 | volume=33 | pages=1–100 | doi=10.2307/1968102 | jstor=1968102 | issue=1 | journal=[[Annals of Mathematics]] | jfm=58.0226.02 | mr=1503035| zbl=0004.05905}}
</ref> The central theorem can now be proved by [[Banach algebra]] methods, and contains much, though not all, of the previous theory.

== See also ==
* [[Wiener's Tauberian theorem]]
* [[Hardy–Littlewood Tauberian theorem]]
* [[Haar's Tauberian theorem]]

==References==
{{reflist}}

==External links==
*{{springer|title=Tauberian theorems|id=T/t092280}}
* {{cite book | last=Korevaar | first=Jacob | title=Tauberian theory. A century of developments | series=Grundlehren der Mathematischen Wissenschaften | volume=329 | publisher=[[Springer-Verlag]] | year=2004 | isbn=978-3-540-21058-0 |pages=xvi+483|mr=2073637|zbl=1056.40002| doi=10.1007/978-3-662-10225-1 }}
* {{cite book | first1=Hugh L.|last1= Montgomery | authorlink=Hugh Montgomery (mathematician) |author2-link=Robert Charles Vaughan (mathematician)|first2=Robert C.|last2= Vaughan | title=Multiplicative number theory I. Classical theory | series=Cambridge Studies in Advanced Mathematics | volume=97 | year=2007 | isbn=978-0-521-84903-6 | pages=147–167 | publisher=[[Cambridge University Press]] | place=Cambridge | mr=2378655|zbl=1142.11001}} 


[[Category:Tauberian theorems]]
[[Category:Series (mathematics)]]
[[Category:Summability methods]]
[[Category:Summability theory]]